ⓘ Normal variancemean mixture
In probability theory and statistics, a normal variancemean mixture with mixing probability density g {\displaystyle g} is the continuous probability distribution of a random variable Y {\displaystyle Y} of the form
Y = α + β V + σ V X, {\displaystyle Y=\alpha +\beta V+\sigma {\sqrt {V}}X,}where α {\displaystyle \alpha }, β {\displaystyle \beta } and σ > 0 {\displaystyle \sigma > 0} are real numbers, and random variables X {\displaystyle X} and V {\displaystyle V} are independent, X {\displaystyle X} is normally distributed with mean zero and variance one, and V {\displaystyle V} is continuously distributed on the positive halfaxis with probability density function g {\displaystyle g}. The conditional distribution of Y {\displaystyle Y} given V {\displaystyle V} is thus a normal distribution with mean α + β V {\displaystyle \alpha +\beta V} and variance σ 2 V {\displaystyle \sigma ^{2}V}. A normal variancemean mixture can be thought of as the distribution of a certain quantity in an inhomogeneous population consisting of many different normal distributed subpopulations. It is the distribution of the position of a Wiener process Brownian motion with drift β {\displaystyle \beta } and infinitesimal variance σ 2 {\displaystyle \sigma ^{2}} observed at a random time point independent of the Wiener process and with probability density function g {\displaystyle g}. An important example of normal variancemean mixtures is the generalised hyperbolic distribution in which the mixing distribution is the generalized inverse Gaussian distribution.
The probability density function of a normal variancemean mixture with mixing probability density g {\displaystyle g} is
f x = ∫ 0 ∞ 1 2 π σ 2 v exp − x − α − β v 2 σ 2 v) g v d v {\displaystyle fx=\int _{0}^{\infty }{\frac {1}{\sqrt {2\pi \sigma ^{2}v}}}\exp \left{\frac {x\alpha \beta v^{2}}{2\sigma ^{2}v}}\right)gv\,dv}and its moment generating function is
M s = exp α s M g β s + 1 2 σ 2 s 2, {\displaystyle Ms=\exp\alpha s\,M_{g}\left\beta s+{\frac {1}{2}}\sigma ^{2}s^{2}\right,}where M g {\displaystyle M_{g}} is the moment generating function of the probability distribution with density function g {\displaystyle g}, i.e.
M g s = E exp s V) = ∫ 0 ∞ exp s v g v d v. {\displaystyle M_{g}s=E\left\expsV\right)=\int _{0}^{\infty }\expsvgv\,dv.} distribution is conjugate to the normal distribution when serving as the mixing distribution in a normal variance  mean mixture Let the prior distribution
 mean distributed according to another Gaussian distribution yields again a Gaussian distribution. Compounding a normal distribution with variance distributed
 root of the variance is useful because for a wrapped normal distribution, it is an estimator of the standard deviation of the underlying normal distribution
 population mean The assumptions underlying a t  test in its simplest form are that X follows a normal distribution with mean μ and variance σ2 n s2 follows
 Josephson et al. limited themselves to considering two normal mixtures with the same component variances and mixing proportions. As a consequence, their proposal
 the mean When estimating a scale parameter, using a trimmed estimator as a robust measures of scale, such as to estimate the population variance or population
 finite  variance assumptions, an extension of Cochran s theorem on the distribution of quadratic forms holds. An elliptical distribution with a zero mean and
 minimum  variance mean for large normal samples which is to say the variance of the median will be 50 greater than the variance of the mean  see asymptotic
 of the variance This transformation may result in better estimates particularly when the distribution of the variance itself may be non normal For many
 outliers and not the configuration of data near the mean The kurtosis of any univariate normal distribution is 3. It is common to compare the kurtosis
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